The Duhem Theorem

Consider a nonreacting system containing specified amounts of its k components that reaches equilibrium at some temperature 7 and pressure P and let l be the number of phases present. What information is needed for the complete specification of this system in terms of both intensive and extensive variables In this case, thus, the complete determination of its equilibrium state, requires - in addition to 7, P, and the composition of each phase - the amounts of each phase. Hence  

The difference between variables and equations, equal to 2, suggests that specification of two variables suffices for the complete determination of the equilibrium state of a multiphase system, provided that the initial amounts of its components are known. This is referred to as the Duhem Theorem. These two variables can be intensive or extensive. Keep in mind, however, that the number of independent intensive variables is determined by the phase rule (see next Example). 

A cylinder-and-piston assembly, containing one kg of steam at P 20 kPa, is placed into a constant temperature (348.15 K) bath and, after some time has 

If we push the piston farther, however, we notice that as long as the two phases exist at equilibrium  

To accomplish this, an additional property - say, the total volume of the mixture - must be specified. This is because, unlike pressure, the total volume can change with the movement of the piston, i.e. volume and temperature are independent variables. 

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We then move to the more complex cases reactions taking place in heterogeneous systems and multiple reactions. To this purpose we consider first the Phase Rule and the Duhem theorem - as they apply to reacting systems - and discuss the methodology for identifying the number of independent reactions for the formation of a given equilibrium mixture. 

The Phase Rule and the Duhem Theorem for Reacting Systems... 

In developing the Duhem theorem for non-reacting systems in Section 12.7, we subtracted the number of equations from the number of vari-... 

The Duhem theorem therefore can be stated in the following general form An equilibrium system, resulting from specified initial amounts of its components, is completely determined if two variables that vary independently at equilibrium are fixed, independently of the number of reactions and phases involved. This theorem is very important in reacting systems, for we are often interested in determining the equilibrium composition of a system of specified initial amounts of reactants at constant temperature and pressure. 

By relating the fluctuations to the reaction coordinate v, de = SNh Eqs. (12.10) and (12.12) imply that if a system is stable to fluctuations in diffusion, it is also stable to fluctuations in chemical reactions, which is called the Duhem-Jougeut theorem (Kondepudi and Prigogine, 1999). However, a nonequilibrium steady state involving chemical reactions may be unstable even if the system is stable with respect to diffusion. 

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17],... 

Derive Eq. (2.107) using this relationship and the Gibbs-Duhem theorem,... 

Vjt is the stoichiometric coefficient, we can obtain (exc. 12.4) condition (12.4.5). Thus a system that is stable to diffusion is also stable to chemical reactions. This is called the Duhem-Jougeut theorem [3, 4] A more detailed discussion of this theorem and many other aspects of stability theory can be found in the literature [2]. 

Since the phase rule treats only the intensive state of a system, it apphes to both closed and open systems. Duhem s theorem, on the other hand, is a nJe relating to closed systems only For any closed system formed initially from given masses of preseribed ehemieal speeies, the equilibrium state is completely determined by any two propeities of the system, provided only that the two propeities are independently variable at the equilibrium state The meaning of eom-pletely determined is that both the intensive and extensive states of the system are fixed not only are T, P, and the phase compositions established, but so also are the masses of the phases. 

The derivation of the phase rule is based upon an elementary theorem of algebra. This theorem states that the number of variables to which arbitrary values can be assigned for any set of variables related by a set of simultaneous, independent equations is equal to the difference between the number of variables and the number of equations. Consider a heterogenous system having P phases and composed of C components. We have one Gibbs-Duhem equation of each phase, so we have the set of equations... 

This statement is similar to Duhem s theorem, which states that values must be assigned to only two independent variables in order to define the state of a closed system for which the original number of moles of each component is known. 

Thus, another approach consists in selecting some boundary conditions and properties. It is obvious that all exact correlation functions must satisfy and incorporate them in the closure expressions at the outset, so that the resulting correlations and properties are consistent with these criteria. These criteria have to include the class of Zero-Separation Theorems (ZSTs) [71,72] on the cavity function v(r), the indirect correlation function y(r) and the bridge function B(r) at zero separation (r = 0). As will be seen, this concept is necessary to treat various problems for open systems, such as phase equilibria. For example, the calculation of the excess chemical potential fi(iex is much more difficult to achieve than the calculation of usual thermodynamic properties since one of the constraints it has to satisfy is the Gibbs-Duhem relation... 

Duhem s theorem states that, for any closed system formed initially given masses of particular chemical species, the equilibrium state is compl determined (extensive as well as intensive properties) by specification of any independent variables. This theorem was developed in Sec. 12.2 for nonrea systems. It was shown there that the difference between the number of indepet] variables that completely determine the state of the system and the number independent equations that can be written connecting these variables is... 

If chemical reactions occur, then we must introduce a new variable, the i coordinate e for each independent reaction, in order to formulate the mate balance equations. Furthermore, we are able to write a new equilibrium rela [Eq. (15.8)] for each independent reaction. Therefore, when chemical-rea equilibrium is superimposed on phase equilibrium, r new variables appear r new equations can be written. The difference between the number of va and number of equations therefore is unchanged, and Duhem s theorem originally stated holds for reacting systems as well as for nonreacting syste Most chemical-reaction equilibrium problems are so posed that it is 1 theorem that makes them determinate. The usual problem is to find the corn-tion of a system that reaches equilibrium from an initial state of fixed an of reacting species when the two variables T and P are specified. 

Duhem s theorem is another rule, similar to the phase rule, but less celebratec It applies to closed systems for which the extensive state as well as the intensiv state of the system is fixed. The state of such a system is said to be completel determined and is characterized not only by the 2 + (iV—l)ir intensive phase rule variables but also by the it extensive variables represented by the masse (or mole numbers) of the phases. Thus the total number of variables is... 

On the basis of this result, Duhem s theorem is stated as follows ... 

The complete investigation of this important theorem (usually attributed to Lejeune Dirichlet) is difficult see Duhem, Lemons sur l lectricit6 et le Magn6tisme, Paris, 1891, 1, 159 Maxwell, Treatise on Electricity and Magnetism, Oxford, 1892, 1, 136. 

Spear F. S. (1988) The Gibbs method and Duhem s theorem the quantitative relationships among P, T, chemical potential, phase composition and reaction progress in igneous and metamorphic systems. Contrib. Mineral. Petrol 99, 249-256. 

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